In Atmospheric observational studies many statistical concepts and calculations are involved, for example, variance, spectrum, correlation, EOF (Empirical Orthogonal Function), EEOF (Extended Empirical Orthogonal Function), etc. Some statistical calculations are quite simple, while others may be lengthy and expensive. In many cases, use of the frequency domain can result in improved efficiency for statistical calculations with time-filtered data. Only the Fourier coefficients within the filtered frequency band are significantly nonzero; moreover, these coefficients contain all the information of interest. This paper discusses several statistical calculations based on Fourier coefficients and presents the formulas needed to effect these calculations:
1) A method is given which allows an estimate of the degrees of temporal freedom of two correlated time series, utilizing the frequency spectra of these two time series.
2) A simple way to perform seasonal analyses is proposed which employs a half-year summer/winter projection operator in the frequency domain.
3) A modified lag-correlation calculation is suggested from which lag correlations may be easily obtained in the frequency domain.
4) A spectral approach is presented for EOF and EEOF analyses which reduces the size of the matrix to be solved in the eigen problem. No matter how many grid points are covered by the analyzed area or how many lag steps are involved, the size of the matrix to be solved is always s x s where s is the number of Fourier coefficients in the filter bandpass, resulting in a significant reduction in computation time.
Some of these calculation methods have been used in the study of low frequency oscillations in the large-scale stratospheric temperature field (Gao and Stanford 1988).